Treating units algebraically and dimensional analysis

We have seen multiple times in our life

that distance can be viewed as rate times
time.

Now what I wanna do in this video is use
this fairly

simple formula right over here, this
fairly simple equation, to understand that

units can really be viewed as algebraic
objects, that you can kind

of treat them like variables as we work
through a formula or equation.

Which could be really, really helpful to
make sure that

our, that we're getting results in units
that actually make sense.

So for example, if someone were to give
you a rate, if they were to say a

rate of, let's say, 5 meters

per second, and they were to give you a
time, a time of 10

seconds, then we can pretty, in a pretty
straight forward way, apply this formula.

We say well distance is equal to our rate,
5 meters per

second, times our time, times our time
which is 10 seconds.

And what's neat here is we can treat the
units,

as I just said, like algebraic constructs,
kind of like variables.

So this would be equal to, well

multiplication doesn't matter what order
we multiply in.

So we can change them, the order.

This is the same thing as 5 times 10.

5 times 10 times meters per second, times
meters per seconds, times seconds.

And if we were to treat our units as these

kinds of algebraic objects, and we could
say, hey look,

we have seconds divided by seconds, or you
have a

seconds in the denominator multiplied by
seconds in the numerator.

Those are going to cancel out.

And 5 times 10, of course is, 5 times 10,
of course is 50.

So we would be left with 50, and the

units that we're left with are the meters,
50 meters.

So that's pretty neat.

The units worked out.

When we treated the units out like
algebraic objects, they worked out so that

our end units for distance were in meters,
which is a unit of distance.

Now you're saying, okay, that's, that's
cute and

everything, but this seems like a little
bit

of too much overhead to worry about when
I'm just doing a simple formula like this,

but what I wanna show is that even with a
simple, with a simple formula like

distance is equal to rate times time, what
I just did could actually be quite useful.

And this thing that I'm doing is actually
called dimensional analysis.

And it's useful for something as simple as

distance equals rate times time, but as
you go

into physics and chemistry and
engineering, you'll see

much, much, much more I would say hairy
formulas.

And when you do the dimensional analysis,
it makes

sure that your, that the math is working
out right.

It makes sure that you're getting the
right units.

But even with this, let's try a slightly
more complicated example.

Let's say that our rate is, let's say,

let's keep our rate at 5 meters per
second.

But let's say that someone gave us the
time.

Instead of giving it in seconds, they give
it in hours.

So they say the time is equal to 1 hour.

So now let's try to apply this formula.

So we're gonna get distance is equal to 5
meters per second, 5

meters per second, times time, which is 1
hour, times 1 hour.

But what's that gonna give us?

Well the 5 times the 1, so we multiply the
5 times the 1.

That's just going to give us 5.

Well then we have to, remember we have to,
the units algebraically.

Where we're going to do our dimensional
analysis.

So it's 5, that we have meters per second,
times hours, times hours.

Or you could say 5 meter hours per second.

Well this doesn't look like a, this isn't
a, a,

a set of units that we know, that, that
makes

sense to us, this doesn't feel like our
traditional units

of distance, so we wanna cancel this out
in some way.

And it might jump out of you, well if, if
we can

get rid of this hours, if we could express
it in terms

of seconds, then that would cancel here
and we'd be left with

just the meters, which is a unit of
distance that we're familiar with.

So how do we do that?

Well we'd wanna multiply this thing by
something that has hours in

the denominator, and seconds in the

numerator, times essentially, seconds per
hour.

Well how many seconds are there per hour?

Well, there are 3600, let me do this in a,
I'll do this color, there are 3600 seconds

per hour, or you could even say that there
are 3600 seconds for every 1 hour.

So when you, now when you multiply, these
hours will

cancel with these hours, these seconds
will cancel with those seconds,

and we are left with, we are left with 5
times

3600, what is that, that's 5 times 3000
would be 15000.

5 times 600 is another 3000.

So that is, it's equal to 18000, and the
only units that

we're left with, we just have the meters
there 18 oh, it's

18000, 18000, 18000 meters, right?

And so, this is, we're done.

We've now expressed our distance in terms
of units that we recognize.

If you go 5 meters per second, for 1 hour,
you will go 18000 meters.

But let's just use our little dimensional
analysis muscles a little bit more.

What if, what if we didn't want the answer

in meters, but we wanted the answer in
kilometers?

What could we do?

Well we could take that 18000 meters,
18000 meters, and if

we could multiply it by something that has
meters in the

denominator, meters in the denominator,
and kilometers in the numerator, then

these meters would cancel out and we'd be
left with the kilometers.

So what could we multiply it so we're not
really changing the value?

Well we want to multiply it by essentially
1, so

we wanna write equivalent things in the
numerator and the denominator.

So 1 kilometer is equivalent to 1000
meters.

So one way to think about it, we're just
multiplying this thing by 1.

One kilometer over 1000 meters.

Well, one kilometer is 1000 meters.

So this thing is equivalent to 1.

But what's in need is when you multiply,
we

have meters cancelling with meters, until
you're left with

18000 divided by 1000 is equal to 18, and

then the only units we're left with is the
kilometers.

And we are done, we have re-expressed our

distance instead of in meters, in terms of
kilometers.